Azure Container Registry is a private registry for hosting container images, which is hosted for you on Azure platform. You can use it to store your private Docker images for all types of container deployments. Azure Container Registry integrates well with orchestrators hosted in Azure Container Service, including Docker Swarm, DC/OS, and Kubernetes. It is fully compatible with pen source Docker Registry v2.

Such registries are mostly used in following scenarios:

Store and manage container images across all types of Azure deployments

Use standard open source command line interface (CLI) tools

Simplify registry access with Azure Active Directory

Maintain Windows and Linux container images in a single Docker registry

Once the container registry is created you will see following:

To test repository create an ASP.NET Core WebApi project with docker support.
If you don't know how to do that, please take a look here.

After the project is created, you will see following:

If you run this project (F5), navigate to URL http://localhost:port/api/values
and you will get some result. This means your API is working locally.
After you have build the project docker compose several times go to command prompt and execute

docker images

You will notice following image:

webapi.docker dev 2d504b16f8f1 7 hours ago 299MB

This image is created by Visual Studio build process. To deploy this image Azure Registry go to WebApi.Docker project, right-mouse click and publish the project directly to registry. Go to azure portal and notice published repository.

After few deployments, take a look in you command prompt for images created after each deployment: